One can easily prove, that in finite dimensional vector space $V$ for every subspace $U$ there always exists a direct complement. It seems to me that it is not the case in infinite dimensional vector spaces.
I wonder if there is any example of some infinite dimensional vector space $V$ (normed) and its subspace $U$ that $U$ doesn't have a direct complement?
I want to emphasize that I use the following definition of subspace:
Subspace is a linear topological space.
Theorem which comes to mind:
Let $H$ be a Hilbert space and $W$ its closed subspace. Than there exists an orthogonal complement to W, i. e. $H = W \oplus W^{\perp}.$
So it seems like even in Hilbert space the subspace should be closed to have a direct complement.
